Suppose G is a nilpotent, finite group. We show that if {a,b} is any
2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b)
has a hamiltonian path. This implies there is a hamiltonian path in every
connected Cayley graph on G that has valence at most 4.Comment: 7 pages, no figures; corrected a few typographical error

Morris, Dave Witte

English

arXiv

Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies that every connected, cubic Cayley graph on G has a hamiltonian path

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 7, Number 1, Pages 41–47ISSN 1715-08682-GENERATED CAYLEY DIGRAPHS ON NILPOTENTGROUPS HAVE HAMILTONIAN PATHSDAVE WITTE MORRISAbstract. Suppose G is a nilpotent, finite group. We show that if{a, b} is any 2-element generating set of G, then the corresponding Cay-ley digraph−−→Cay(G; a, b) has a hamiltonian path. This implies that allof the connected Cayley graphs of valence ≤ 4 on G have hamiltonianpaths.1. IntroductionLet G be a group. (All groups are assumed to be finite.)Definition. For any subset S of G, the Cayley digraph of S on G is thedirected graph whose vertices are the elements of G, and with a directededge g → gs, for every g ∈ G and s ∈ S. It is denoted −−→Cay(G;S).It is known that every connected Cayley digraph on G has a hamiltonianpath if either G is abelian (see Lemma 2.1) or G has prime-power order (seeTheorem 2.2). Since abelian groups and p-groups are the basic examples ofnilpotent groups, it is natural to ask whether it suffices to assume that G isnilpotent. We provide some evidence that this may indeed be the case:Theorem 1.1. Every connected Cayley digraph of outvalence 2 on anynilpotent group has a hamiltonian path.There is no need to make any restriction on the outvalence of−−→Cay(G;S)if we assume the nilpotent group G has only one Sylow subgroup that is notabelian:Theorem 1.2. If G = P × A, where P has prime-power order, and A isabelian, then every connected Cayley digraph on G has a hamiltonian path.Remark. In abstract terms, the assumption G = P × A in Theorem 1.2 isequivalent to assuming that G is nilpotent and the commutator subgroupof G has prime-power order.Received by the editors April 5, 2011, and in revised form June 30, 2011.2000 Mathematics Subject Classification. 05C20, 05C25, 05C45, 20D15.Key words and phrases. Cayley digraph, hamiltonian path, nilpotent group, Cayleygraph.c©2012 University of Calgary4142 DAVE WITTE MORRISThe above results for directed graphs have the following consequence for(undirected) Cayley graphs.Corollary 1.3. Every connected Cayley graph of valence ≤ 4 on any nilpo-tent group has a hamiltonian path.Remark. One can show quite easily that if S = {a, b} is a 2-element gen-erating set of a group G, such that |a| = 2, |b| = 3, and |G| > 9|ab2|,then−−→Cay(G; a, b) does not have a hamiltonian path. (This observation isattributed to J. Milnor [2, p. 267].) Examples in which G is (super)solvablecan be constructed by taking G to be an appropriate semidirect productZ6 n Zp, where p is a large prime that is congruent to 1 modulo 6. There-fore, the word “nilpotent” cannot be replaced with the word “solvable” (oreven “supersolvable”) in the statement of Theorem 1.1.After some preliminaries in Section 2, the above results are proved inSection 3, by using the methods of [5]. See the bibliography of [4] for ref-erences on the search for hamiltonian cycles in Cayley graphs on general(non-nilpotent) groups.2. PreliminariesAll groups in this paper are assumed to be finite.Notation. Let G be a group, let S be any subset of G, and let H be anysubgroup of G.• The Cayley digraph −−→Cay(G;S) is the directed graph whose vertexset is G, and with an arc from g to gs, for every g ∈ G and s ∈ S.• The Cayley graph Cay(G;S) is the (undirected) graph that underlies−−→Cay(G;S). Thus, its vertex set is G, and g is adjacent to both gsand gs−1, for every g ∈ G and s ∈ S.• H\−−→Cay(G;S) denotes the digraph in which:◦ the vertices are the right cosets of H, and◦ there is a directed edge from Hg to Hgs, for each g ∈ G ands ∈ S.• HG = 〈g−1hg | h ∈ H, g ∈ G〉 is the normal closure of H in G.• 〈S−1S〉 = 〈 s−11 s2 | s1, s2 ∈ S 〉 is the arc-forcing subgroup. Notethat, for any a ∈ S, we have 〈S−1S〉 = 〈a−1S〉 = 〈 a−1s | s ∈ S 〉.• For s1, s2, . . . , sn ∈ S, we use (si)ni=1 = (s1, s2, s3, . . . , sn) to denotethe walk in−−→Cay(G;S) that visits (in order) the verticese, s1, s1s2, s1s2s3, . . . , s1s2 · · · sn.Also, (s1, s2, s3, . . . , sn)# denotes the walk (s1, s2, s3, . . . , sn−1) thatis obtained by deleting the last term of the sequence.2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS 43Terminology. Contrary to most authors, we consider both K2 and the loopon a single vertex to have hamiltonian cycles. This is because each of thesegraphs has a hamiltonian path whose terminal vertex is adjacent to its initialvertex.The following well-known observation is very easy to prove.Lemma 2.1 ([1, Thm. 30.3, p. 506]). Every connected Cayley digraph onany abelian group has a hamiltonian path.In the remainder of this section, we recall some useful results from [5].Theorem 2.2 (Witte [5]). Every nontrivial, connected Cayley digraph onany group of prime-power order has a hamiltonian cycle.Lemma 2.3 (cf. [5, Lem. 4.1]). Suppose H is a subgroup of a group G. IfG is nilpotent, then there is a subnormal seriesH = H1 / H2 / · · · / Hm = HGof HG, such that, for 1 ≤ k < m:(i) Hk+1/Hk is generated by a G-conjugate of H, and(ii) Hk+1/Hk has prime-power order.Proof. The desired conclusion is proved in [5, Lem. 4.1] under the strongerassumption that G is a p-group. The general result follows from this specialcase, since every nilpotent group is a direct product of p-groups.For the reader’s convenience, we provide a proof from scratch: givenH1, . . . ,Hk, with Hk ( HG, we show how to construct Hk+1. Since H ⊆Hk ( HG, we have Hk 6/ G, which means NG(Hk) 6= G. Then, becauseproper subgroups of nilpotent groups are never self-normalizing [3, Haupt-satz III.2.3(c), p. 260], we have NG(Hk) ( NG(NG(Hk)), so we may choosesome x ∈ G, such thatx normalizes NG(Hk), but x /∈ NG(Hk).Since x /∈ NG(Hk), we know x−1Hkx 6⊆ Hk. We also know (from property (i)and induction) that Hk is generated by G-conjugates of H. Hence, thereexists g ∈ G, such thatg−1Hg 6⊆ Hk, and g−1Hg ⊆ x−1Hkx.Let Hk+1 = Hk · (g−1Hg). Then:• Hk+1 properly contains H, because g−1Hg 6⊆ Hk, and• Hk/Hk+1, because g−1Hg ⊆ x−1Hkx ⊆ NG(Hk) (since x normalizesNG(Hk)).By construction, the quotient Hk+1/Hk is generated by g−1Hg.Since G is nilpotent, it is the direct product of its Sylow subgroups: G =P1×· · ·×Pr. Hence, we may write g = g1g2 · · · gr, with gi ∈ Pi. Furthermore,every subgroup of G is the direct product of its intersections with the Sylow44 DAVE WITTE MORRISsubgroups of G. Therefore, since g−1Hg 6⊆ Hk, there is some i, such that(g−1Hg) ∩ Pi 6⊆ Hk ∩ Pi. This means g−1i Hgi 6⊆ Hk. We also haveg−1i Hgi ⊆〈H, (g−1i Hgi) ∩ Pi〉 ⊆ 〈H, g−1Hg〉 ⊆ 〈Hk, Hk+1〉 = Hk+1.Hence, there is no harm in assuming g = gi ∈ Pi. Then g−1Hg ⊆ PiH, so|Hk+1/Hk| is a divisor of |Pi|, which is a prime-power. Remark 2.4. The assumption that G is nilpotent in Lemma 2.3 can bereplaced with the assumption that HG is a p-group (for some prime p).To see this, note that if Hk 6/ HG, then, since HG is nilpotent, the proofof Lemma 2.3 constructs an appropriate subgroup Hk+1 = Hk · (g−1Hg),with g ∈ HG. On the other hand, if Hk / HG, then every G-conjugate of Hnormalizes Hk, so it is easy to construct Hk+1. (Since |Hk+1/Hk| is a divisorof |HG|, which is a power of p, property (ii) is automatically satisfied.)Lemma 2.5 ([5, Lem. 5.1]). Suppose S generates a group G, and let H+and H− be subgroups of G with H− / H+. If• there is a hamiltonian cycle in H+\−−→Cay(G;S),• every connected Cayley digraph on H+/H− has a hamiltonian cycle,and• H+/H− is generated by a G-conjugate of the arc-forcing subgroup〈S−1S〉,then there is a hamiltonian cycle in H−\−−→Cay(G;S).Essentially the same proof establishes an analogous result for hamiltonianpaths, but we need only the following simplified version in which H− istrivial:Lemma 2.6 (cf. [5, Lem. 5.1]). Suppose S generates a group G, and letH = 〈S−1S〉 be the arc-forcing subgroup. If• there is a hamiltonian cycle in H\−−→Cay(G;S), and• every connected Cayley digraph on H has a hamiltonian path,then there is a hamiltonian path in−−→Cay(G;S).Proof. Let• (si)mi=1 be a hamiltonian cycle in the quotient digraph H\−−→Cay(G;S),and• (s1s2 · · · sm−1ai)n−1i=1 be a hamiltonian path in the Cayley digraph−−→Cay(H; s1s2 · · · sm−1S).Then it is not difficult to verify that(s1, s2, . . . , sm−1, ai)ni=1#is a hamiltonian path in−−→Cay(G;S). 2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS 453. Proofs of the main resultsThe heart of our argument is contained in the following result, which isadapted from the proof of [5, Thm. 6.1], and may be of independent interest.Proposition 3.1. Let• S be a generating set of a finite group G, and• H = 〈S−1S〉 be the arc-forcing subgroup.If• G is nilpotent, and• every connected Cayley digraph on H has a hamiltonian path (orhamiltonian cycle, respectively),then−−→Cay(G;S) has a hamiltonian path (or hamiltonian cycle, respectively).Proof. Consider the subnormal seriesH = H1 / H2 / · · · / Hm = HGthat is provided by Lemma 2.3, and choose some a ∈ S. Since〈a,H〉 = 〈a, a−1S〉 = 〈S〉 = G,we know that a generates the quotient group G/HG = G/Hm. Thus,Hm\−−→Cay(G; a) is a directed cycle. Furthermore, for each k, Theorem 2.2tells us that every connected Cayley digraph on Hk+1/Hk has a hamilton-ian cycle. Thus, repeated application of Lemma 2.5 (with H+ = Hk+1 andH− = Hk, for k = m − 1,m − 2, . . . , 1) tells us that H1\−−→Cay(G;S) has ahamiltonian cycle. Since H1 = H, this means H\−−→Cay(G;S) has a hamilton-ian cycle.By assumption, we also know that every connected Cayley digraph on Hhas a hamiltonian path (or hamiltonian cycle, respectively). Therefore,Lemma 2.6 (or Lemma 2.5 with H+ = H and H− = {e}) provides a hamil-tonian path (or hamiltonian cycle) in−−→Cay(G;S). Remark. The proof of Theorem 2.2 is a minor modification of the proof ofProposition 3.1. Namely, rather than appealing to Theorem 2.2 in order toknow that every connected Cayley digraph on Hk+1/Hk has a hamiltoniancycle, one can assume this is true by induction on |G|. The same inductionhypothesis also implies that every connected Cayley digraph on H has ahamiltonian cycle.Corollary 3.2. Let S be a generating set of the group G. If• G is nilpotent, and• the arc-forcing subgroup H = 〈S−1S〉 is abelian,then−−→Cay(G;S) has a hamiltonian path.Proof. Lemma 2.1 tells us that every connected Cayley digraph on H has ahamiltonian path, so Proposition 3.1 applies. 46 DAVE WITTE MORRISProof of Theorem 1.1. Let {a, b} be a 2-element generating set for G.Then the arc-forcing subgroup H = 〈a−1b〉 is cyclic, so it is abelian. There-fore Corollary 3.2 provides a hamiltonian path in−−→Cay(G; a, b). Proof of Theorem 1.2. Let−−→Cay(G;S) be a connected Cayley digraph onG = P × A, and let H = 〈S−1S〉 be the arc-forcing subgroup. We mayassume the generating set S is minimal.Case 1. Assume H 6= G. By induction on |G|, we may assume every con-nected Cayley digraph on H has a hamiltonian path. Then Proposition 3.1provides a hamiltonian path in−−→Cay(G;S).Case 2. Assume H = G. Choose some a ∈ S, and let : G → P be thenatural projection homomorphism. Since G = H = 〈a−1S − {e}〉, and theminimal generating sets of any finite p-group all have the same cardinality[3, Satz III.3.15, p. 273], there is a proper subset S0 of S, such that 〈S0〉 = P .Since G/P ∼= A is abelian, this implies [G,G] ⊆ 〈S0〉. Therefore 〈S0〉 E G.Let N = 〈S0〉 E G. Since S0 is a proper subset of S, and S is minimal,we know N 6= G. Also, we may assume [G,G] is nontrivial, for otherwiseLemma 2.1 provides a hamiltonian path in−−→Cay(G;S). Therefore N is non-trivial. Hence, by induction on |G|, we may assume every connected Cayleydigraph on N or G/N has a hamiltonian path; let• (si)ni=1 be a hamiltonian path in−−→Cay(N ;S0), and• (tj)qj=1 be a hamiltonian path in−−→Cay(G/N ;S).Then it is easy to see (and well known) that((si)ni=1, tj)q+1j=1# is a hamilton-ian path in−−→Cay(G;S). Proof of Corollary 1.3. Suppose Cay(G;S) is a connected Cayley graphof valence ≤ 4, and G is nilpotent. There is no harm in assuming that thegenerating set S is minimal. Let S2 be the set of elements of order 2 in S.Also, let P be the Sylow 2-subgroup of G, so G = P ×K, where |K| is odd.If #S − #S2 ≤ 1, then, since S2 ⊆ P , we know K ∼= G/P is cyclic.Therefore K is abelian, so Theorem 1.2 applies.We may now assume #S −#S2 ≥ 2. Then4 ≥ valence of Cay(G;S) = #(S ∪ S−1)= 2(#S −#S2) + #S2 ≥ 2(#S −#S2) ≥ 2 · 2.We must have equality throughout, so #S = 2 (and S2 = ∅). Then Theo-rem 1.1 provides a hamiltonian path in Cay(G;S). The following generalization of Theorem 2.2 is sometimes useful.Corollary 3.3. Suppose• S is a nonempty generating set of a group G,• N is a normal p-subgroup of G, for some prime p, and• there exists a ∈ G, such that S ⊂ aN .2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS 47Then−−→Cay(G;S) has a hamiltonian cycle.Proof. Let H = 〈S−1S〉 ⊂ (aN)−1(aN) = N . Since N / G, this impliesHG ⊂ N , so HG is a p-group. Hence, Remark 2.4 provides a subnormalseries as in Lemma 2.3, and Theorem 2.2 tells us that every connected Cayleydigraph on H has a hamiltonian cycle. Then the proof of Proposition 3.1provides a hamiltonian cycle in−−→Cay(G;S). AcknowledgmentsThis research was carried out during a visit to the University of WesternAustralia. I enthusiastically thank that university, particularly the mem-bers of the School of Mathematics and Statistics, for their warm hospitalitythat made my visit both productive and enjoyable. The work was partiallysupported by a grant from the Natural Sciences and Engineering ResearchCouncil of Canada and by funds from Australian Research Council Federa-tion Fellowship FF0770915.References1. J. A. Gallian, Contemporary Abstract Algebra, 7th ed., Houghton Mifflin, Boston, 2010.2. W. Holsztyn´ski and R. F. E. Strube, Paths and circuits in finite groups, Discrete Math.22 (1978), no. 3, 263–272.3. B. Huppert, Endliche Gruppen I., Springer, New York, 1967.4. K. Kutnar, D. Marusˇicˇ, J. Morris, D. W. Morris, and P. Sˇparl, Hamiltonian cycles inCayley graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27–71.5. D. Witte, Cayley digraphs of prime-power order are hamiltonian, J. Comb. Th. B 40(1986), 107–112.Department of Mathematics and Computer Science,University of Lethbridge, Lethbridge, Alberta, T1K 3M4, CanadaE-mail address: Dave.Morris@uleth.ca

2-generated Cayley digraphs on nilpotent groups have hamiltonian paths

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